Báo cáo y học: "Cancer proliferation and therapy: the Warburg effect and quantum metabolism" pot

ualberta.ca 3 Department of Experimental Oncology, Cross Cancer Research Institute, Edmonton, Alberta, Canada Abstract Background: Most cancer cells, in contrast to normal differentiated

R E S E A R C H Open Access

Cancer proliferation and therapy: the Warburg

effect and quantum metabolism

Lloyd A Demetrius1, Johannes F Coy2, Jack A Tuszynski3*

* Correspondence: jtus@phys.

ualberta.ca

3 Department of Experimental

Oncology, Cross Cancer Research

Institute, Edmonton, Alberta,

Canada

Abstract

Background: Most cancer cells, in contrast to normal differentiated cells, rely on aerobic glycolysis instead of oxidative phosphorylation to generate metabolic energy,

a phenomenon called the Warburg effect

Model: Quantum metabolism is an analytic theory of metabolic regulation which exploits the methodology of quantum mechanics to derive allometric rules relating cellular metabolic rate and cell size This theory explains differences in the metabolic rates of cells utilizing OxPhos and cells utilizing glycolysis This article appeals to an analytic relation between metabolic rate and evolutionary entropy - a demographic measure of Darwinian fitness - in order to: (a) provide an evolutionary rationale for the Warburg effect, and (b) propose methods based on entropic principles of natural selection for regulating the incidence of OxPhos and glycolysis in cancer cells

Conclusion: The regulatory interventions proposed on the basis of quantum metabolism have applications in therapeutic strategies to combat cancer These procedures, based on metabolic regulation, are non-invasive, and complement the standard therapeutic methods involving radiation and chemotherapy

Background

Cancer is an age-dependent disease characterized by five key hallmarks in cell physiol-ogy that drive the progressive change of normal differentiated cells into diverse states

of malignancy [1]: autonomous growth-replication in the absence of growth signals; insensitivity to anti-growth signals; apoptosis-evasion of programmed cell death, angio-genesis-the induction of the growth of new blood vessels; invasion and metastasis The age-dependency of cancer [2] and the relatively rare incidence of the disease during an average human life time suggest that adaptive mechanisms exist in cells and tissues to prevent this multi-step transition from a normal differentiated cell into malignancy Consequently, each of these physiological changes constitutes the rupture

of anti-tumor defences developed during the evolutionary history of the organism It may be worth observing that a graph of the logarithm of the total cancer incidence against age approximates to a straight line with a gradient of 6-7 (the value of the power-law exponent) suggesting that 6-7 separate events are required for neoplastic transformation of a‘typical’ human cell [3]

Cancer cells may be considered as autonomous units which have an impaired capa-city to maintain the metabolic stability of the organism in which they reside Anti-cancer therapies are corrective measures designed to remedy this impairment

© 2010 Demetrius et al; licensee BioMed Central Ltd This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and

by eliminating the errant cells.The nature of these corrective measures has undergone

a significant development, starting with surgical resection of solid tumors followed by

radiation and then chemotherapy, as the understanding of the biology of cancer has

increased The non-surgical therapies which came to dominate the treatment of the

disease were based on the proposition that the disease is primarily the result of

dynamic changes in the genome This gene oriented perspective led to the notion that

decoding the genetic instruction that determines the cancer phenotype would elucidate

its origin and thus provide an effective biological basis for therapy Genes represent

the blueprint for phenotypic expression Accordingly, the genetic model entailed

thera-pies based on the complete elimination of cancer cells Radiation and chemotherapy

were the first class of anti-cancer strategies which this model invoked These two

ther-apeutic methods were designed to eliminate cancer cells from tissues However, due to

the low selectivity of this approach, non-cancer cells are also killed or damaged leading

to severe side effects

The next generation of cancer drugs which developed from this genomic viewpoint explicitly recognized the multi-step progression towards malignancy, and that in most

instances death only occurs when the metastatic state is attained Metastasis is often

triggered by angiogenesis, the proliferation of a network of blood vessels that

pene-trates into cancerous tissue, supplying nutrients and oxygen [4]

Drugs that impede the formation of tumor blood vessels were therefore proposed as therapeutic agents in the combat against malignancy [5] There exist, however, some

disadvantages to this mode of therapy as angiogenic inhibitors sometimes trigger side

effects and induce a more invasive type of tumor [6]

Studies in recent years have led to a re-evaluation of the genomic model of cancer and the development of a model based on cell metabolism [7] The research which

triggered this shift from genes to metabolic reactions was done in 1924 by Warburg

[8] who recognized certain critical differences between energy regulation in normal

dif-ferentiated cells and cancer cells Warburg analyzed the ratio of oxidative

phosphoryla-tion (OxPhos) to glycolysis in different tissues of cancer cells and normal cells

Glycolysis under aerobic conditions was found to be particularly high in aggressive

tumors when compared with benign tumors and normal tissues These observations

led Warburg to propose deficiency in OxPhos and elevated glycolysis as the primary

cause of cancer

The discovery of the double helix by Watson and Crick in 1953 and its implications for the understanding of molecular processes in biology diverted interest away from

research into the significance of Warburg’s metabolic hypothesis However, the failure

of the genomic approach to provide effective therapies for certain types of aggressive

cancer, and recent studies [7] creating a rapprochement of genetic and metabolic views

have revived interest in the Warburg hypothesis

The hypothesis, in its simplest form, asserts that cancer is primarily a disease of meta-bolic dysregulation: a switch, inducible by various agents- genetic, nutritional and

environ-mental, from an OxPhos pathway to a glycolytic mode of energy processing This focus on

metabolism as the primary cause for the progressive transition from normalcy to

malig-nancy suggests a radically new approach to cancer therapy The focus is to influence

meta-bolic regulation in cancer cells so that the autonomy, proliferative capacity, invasiveness

and metastasis which define the aggressive cancer phenotype, is never attained

A therapeutic strategy which emphasizes containment rather than annihilation is

a radical departure from methods designed to eradicate cancer cells completely from

tissues Therapies based on metabolic interventions involve two complementary

programs: the down-regulation of glycolysis and the up-regulation of OxPhos

According to Warburg the aggressiveness of a tumor derives from the elevation of the glycolytic mode of energy processing This elevation is considered to be the result

of competition between cells utilizing the glycolytic mode, and cells adopting OxPhos

Hence, the principle that underlies these complementary programs of disease control

is Darwinian: the modulation of the selective advantage of cells using glycolysis and

OxPhos, respectively

It is well known that ATP generation through glycolysis is less efficient than through mitochondrial respiration Hence a long-standing paradox is how cancer cells with

their metabolic disadvantage can survive the competition with normal cells In terms

of biochemical reactions [9], mitochondrial respiration defects lead to activation of the

Akt survival pathway through a mechanism mediated by NADH Respiration-deficient

cells harboring mitochondrial defects exhibit dependency on glycolysis, increased

NADH, and activation of Akt, leading to survival advantage and also drug resistance in

hypoxia [9]

The efforts to implement these therapeutic programs have generated certain signifi-cant questions regarding the analytical characterization of Darwinian fitness, the

capa-city of a cell type to displace related types in competition for resources The problem

can be formulated as follows: (a) What class of physiological, biochemical and

biophy-sical properties of cells confers a selective advantage during evolution of the cancer

phenotype? (b) To what extent can these properties be analytically described in terms

of bio-energetic and kinetic variables associated with the regulatory circuits that

describe the metabolic networks?

These questions are consistent with the view that cancer development proceeds according to an evolutionary process in which a succession of genetic and epigenetic

changes, due to the selective advantage conferred, leads to the progressive

transforma-tion of normal cells to cancer cells [10-12] The resolutransforma-tion of the problems addressed

in (a) and (b) would evidently provide an analytic framework for cancer therapy based

on containment rather than eradication The analysis would also yield a rationale

based on natural selection for Warburg’s hypothesis, and consequently, an evolutionary

understanding of the origin of cancer

The analytical framework for a theory of metabolic regulation in cells capable of addressing problems of cellular adaptation and somatic evolution within living

organ-ism was proposed in a series of articles [13,14], and called Quantum Metabolorgan-ism in

view of the quantum mechanics methodology the theory invoked Quantum

Metabo-lism gives a molecular level explanation of certain empirically derived allometric laws

relating metabolic rate with cell size

The allometric rules are of the form [14]:

Here, P is the metabolic rate, the rate of ATP production, W the cell size The dimensionality parameter d in the scaling exponent, b = d/(d+1), describes the number

of degrees of freedom of the enzymes which catalyze the redox reactions within the

energy transducing organelles: mitochondria (in the case of OxPhos), metabolosomes

(in the case of glycolysis) The proportionality constant, a, depends on the mode of

coupling between the electron transport chain and ADP phosphorylation This mode

of coupling is electrical in the case of OxPhos and chemical in the case of glycolysis

A mathematical theory of evolution by natural selection which, for the first time, considered the effect of resource constraints and finite population size on the outcome

of competition between related types, was described in [15] The cornerstone of this

model was the statistical parameter, evolutionary entropy, a measure of the stability of

population numbers, and an index of Darwinian Fitness

In this article we will integrate Quantum Metabolism with certain analytic relations between evolutionary entropy and metabolic rate to show that selective advantage in

cellular evolution is contingent on the resource constraints - its abundance and

distri-bution and predicted by the metabolic rate, the rate at which cells transform resources

into metabolic work Quantum Metabolism predicts that the metabolic rate of cells

uti-lizing OxPhos and cells utiuti-lizing glycolysis will have the same scaling exponents but

will differ in terms of the proportionality constants We will exploit this prediction,

and the characterization of selective advantage in terms of resource constraints

and metabolic rate, to provide an evolutionary rationale for the cancer phenotype The

evolutionary argument rests on differences in the metabolic rate of cells utilizing

OxPhos and glycolytic pathways, respectively Cellular metabolic rate can be influenced

by perturbing the geometry of the metabolic network or the mode of coupling hence

the incidence of OxPhos and glycloysis in normal and cancer cells can be metabolically

regulated We will appeal to these notions of metabolic intervention to propose

anti-cancer strategies based on arresting the transformation from a benign tumor, to a

malignant tissue, characterized by enhanced glycolysis

This article will give a brief account of Quantum Metabolism and the scaling laws for metabolic rate which the new theory derives We discuss the evolutionary

perspec-tive this theory entails, and then apply the new class of models to propose therapies

based on altering the selective advantage of normal and cancer cells during the

transi-tion to the cancer phenotype

Quantum metabolism

Cellular metabolism is the totality of all chemical reactions in cells carried out by an

organism The characteristics of living organisms, such as their growth, the

mainte-nance of their structure and mass transport, depend on the input of energy from the

environment Metabolism designates the series of chemical reactions that transform

substrates such as glucose into cellular building blocks and energy in the form of ATP

A quantitative understanding of the rules which regulate this energy transformation

is critical for a quantitative characterization of the selective advantage associated with

different modes of energy processing

Quantum Metabolism exploits the methodology of the quantum theory of solids, as developed by Einstein and Debye, to derive a class of analytic rules relating metabolic

rate, with cell size The variables which define these rules are bio-energetic parameters,

whose values depend on the phospholipid composition of the bio-membranes, and

enzymatic reaction rates, which depend on the concentration of substrates in metabolic

reactions

The allometric laws of metabolism

Metabolic rate, the rate at which an organism transforms nutrients into thermal energy

and biological work required for sustaining life, is highly dependent on the organism’s

size This observation draws in large part from the experimental studies of Lavoisier

and Laplace who were the first to demonstrate the relationship between combustion

and respiration The systematic empirical study of the relation between metabolic rate,

P, and body size, W, which began with Rubner [16], and was extended later by Kleiber

[17] for non-domesticated mammals, and by Hemmigsen [18] to uni-cells led to a set

of allometric laws relating metabolic rate with body size

Quantum Metabolism exploits the formalism of quantum mechanics to study the dynamics of energy flow in the electron transfer chain in cells, and provides a molecular

level explanation of the empirical rules documented in Kleiber [17] and Hemmingsen [18]

The derivation of Eq (1) is based on the assumption that the transformation of nutrients into thermal energy and biological work involves the inter-conversion of two

forms of energy [19,20]:

(1) The redox potential difference that is the actual redox potential between the donor and acceptor couples in the electron transport chain

(2) The phosphorylation potential for ATP synthesis

The parameter d in the scaling exponent b = d/(d+1), characterizes the number of degrees of freedom of the enzymes which are embedded in the energy transducing

organelles Enzymatic reactions have an intrinsic direction and the enzymes localized

in the organelles have a given orientation The enzymes are subject to oscillations, due

to the redox potential We will assume that these vibrations can be approximated by

harmonic oscillators We will also assume that the enzymatic vibrations are coupled

and inherited by the energy transducing organelles in which the enzymes reside There

exists a diverse body of empirical support for these assumptions Some of this support

can be annotated as follows Experimental studies of the organization of mitochondrial

networks show that these systems can be regarded as coupled oscillators [21]

Synchronisation of metabolic cycles through gene and enzyme regulation within and between cells has been shown to involve co-ordinated transcriptional cycles not only

in cultured yeast but also importantly in mammalian cells [22-24] Tsong and

colla-borators have demonstrated that dynamical processes govern the function of metabolic

enzymes which can capture and transmit energy from oscillating electric fields [25]

involving electro-conformational coupling [26] and electric modulation of membrane

proteins [27]

Our model rests on the hypothesis that the metabolic energy of the cell is character-ized by the coupled oscillations of the energy transducing organelles Consequently,

there are three levels of metabolic organization to be considered: (a) the energy

con-tained in the vibrations at the level of individual enzymes, (b) the coupling of the

enzymes within the organelles, and (c) the coupling of the energy transducing

orga-nelles within the cell

Because the energy which drives the process of metabolic regulation depends on coupling at two distinct levels, the enzymatic level and the level of the energy

transdu-cing organelles, we can assume that the dimensionality parameter d will depend on

physico-chemical properties of the cellular matrix at: (a) the level of the enzymes,

(b) the level of the mitochondria and metabolosomes Hence, we can assume that d, an

index of the number of degrees of freedom of the enzymes aligned in the organelles,

may not satisfy the condition 1<d<3, as in the Debye model, but may vary in principle

between 1 and infinity

Cycle time and metabolic rate

Quantum Metabolism establishes that the mode of energy transfer may either be

quan-tized or classical, contingent on: (a) the cycle time, τ, of the metabolic processes

trans-forming substrates into products within the network of chemical reactions, and (b) the

relaxation time,τ *, which describes the return time of enzyme concentrations to their

steady state condition after a random perturbation

The mean cycle time, τ, is highly dependent on the external resources, their concen-tration and diversity, whereas the relaxation time, τ*, depends on the density of the

metabolic enzymes and the physical properties of cellular medium [14]

The mean cycle time is the mean turnover time of the enzymes in the reaction pro-cess Based on typical data for ion pumps in biological membranes, the range of values

expected for τ is between 10-6

s and 10-3s [27,28] Recently, the proton-driven ATP synthase rotor has been reported to have Ohmic conductance on the order of 10 fS

[29] Hence, we can estimate for this value the number of protons involved in each

cycle to range from 10 at the high frequency limit to 10, 000 at the low-frequency

limit More recent papers provided a novel description of mitochondria as individual

oscillators whose dynamics may obey collective, network properties in terms of

high-amplitude, self-sustained and synchronous oscillations of bio-energetic parameters

under both physiological and patho-physiological conditions [30] This was

demon-strated through the analysis of their power spectrum that exhibits an exponent vastly

different from random behavior Therefore, a proposed description of the metabolic

activity involving mitochondrial proteins as coupled quantum oscillators of the Debye

type appears to be supported by recent observations

According to Quantum Metabolism, the quantized or classical modes of energy transfer are determined by the extreme values assumed by the ratioτ/τ* We have the

following two scenarios [14]:

(a) Whenτ <<τ *, the mode of energy transfer is discrete and the dynamic of energy flow is described by quantum laws In this case d is finite and the scaling exponent, b,

satisfies 1/2 <b < 1

(b) When τ >>τ*, energy transduction is continuous and the dynamic of energy trans-fer is described by classical laws Here, the parameter d is infinite and the scaling

expo-nent b = 1

Proportionality constant and metabolic rate

The proportionality constant, a, is determined by the mechanism - electrical or

chemi-cal - which describes the coupling between electron transfer and ATP assembly The

coupling between the electron transport chain and ADP phosphorylation is electrical,

in the case of OxPhos, and chemical in the case of glycolytic processes The metabolic

rate in cells utilizing each of these processes will differ, the difference being due to the

proportionality constant

(I) Oxidative phosphorylation OxPhos is a mode of energy processing whereby the cell carries out phosphorylation of ADP during the oxygen-dependent transmission of

electrons down the respiratory chain in the relevant membrane This reaction produces

NADH which then fuels OxPhos to maximize ATP production with minimal

production of lactate The coupling between the electron transport chain and ADP

phosphorylation is generated by the flow of protons across the bio-membrane The

proportionality constant a is a function of the bio-energetic parameters, proton

con-ductance C, and the proton motive potentialΔp These parameters will depend on the

phospholipid composition of the membrane [31]

(IIa) Anaerobic glycolysis (glycolysis which is suppressed by oxygen) Anaerobic gly-colysis describes a mode of energy generation when oxygen is supply limited In this

process, cells direct the pyruvate generated by glycolysis away from the mitochondrial

OxPhos pathway This is achieved by generating lactate This process does not occur

within the mitochondria but in the liquid protoplasm of the cell This generation of

lactate allows glycolysis to continue but results in a lowered ATP production rate The

rate-limiting factors for net ATP production are the efficiencies of glucose delivery and

lactate removal

(IIb) Aerobic glycolysis (glycolysis, which is not suppressed by oxygen) Aerobic glyco-lysis refers to the conversion of glucose to pyruvate and then to lactate regardless of

whether oxygen is present or not This process is also localized in the liquid cytoplasm

This property is shared by normal proliferative tissue

The proportionality constant in both aerobic and anerobic glycolysis is a function of the activity of the glycolytic enzymes

We wish to mention that the terms aerobic and anaerobic glycolysis are imprecise, although they have been used extensively in the literature (see for example [12]) The

term aerobic glycolysis is somehow misleading since both types of glycolysis are in fact

anaerobic Warburg used the term aerobic glycolysis to emphasize the important

differ-ence from normal glycolysis that it is not suppressed by oxygen Any type of glycolysis is

always substantially the same pathway It runs much faster in the absence than in the

presence of oxygen because of feedback regulation For example, when oxygen is

avail-able, the highly efficient ADP-phosphorylating system in the energy-transducing

bio-membrane ensures a high ATP/AMP ratio in the cell, which down-regulates the

glycoly-tic enzymes The distinction between aerobic and anaerobic glycolysis is therefore only

one of net flux rate Consequently, some researchers consider the distinction redundant

Glycolysis is not oxygen-dependent; but it is always (partly) suppressed by oxygen

An empirical observation

Quantum Metabolism predicts that the range of values assumed by the scaling

expo-nent, b, b<1 (an allometric relation), b = 1 (an isometry), is highly dependent on the

cycle time,τ, a quantity which varies with the resource flux - an environmentally

regu-lated property This observation indicates that changes in environmental factors can

induce significant changes in the scaling exponent with concomitant changes in

meta-bolic rate Empirical support for this prediction is given in Nakaya et al [32] The

experiments in [32], reported a shift from a scaling law with b = 3/4 to a scaling law

described by b = 1, contingent on changes in the environmental conditions The

exis-tence of such environmental switches is particularly pertinent in applications of

quan-tum metabolism to somatic evolution Such changes in the scaling exponent indicate a

mechanism for regulating the metabolic profile of cells by imposing various

environ-mental constraints

Oxidative phosphorylation and glycolysis: a bio-energetic comparison

Quantum Metabolism predicts that the scaling laws for cells utilizing OxPhos and

glycolysis will be described by similar scaling exponents but different proportionality

constants The proportionality constant is determined by the mode of coupling

between the electron transport chain and ADP phosphorylation In OxPhos coupling

is achieved by a single common intermediate between the oxidation of a variety of

substrates and ATP formation This intermediate is the trans-membrane proton

gra-dient For glycolysis, there is a single set of enzymes for every coupled reaction

The differences in the mode of coupling entail significant differences in the meta-bolic efficiency of cells utilizing OxPhos and cells using glycolysis With glucose as

substrate, OxPhos generates about 17 times more ATP than glycolysis Consequently,

the metabolic rate of OxPhos (the respiration rate) will also be greater than the

meta-bolic rate of glycolysis (the fermentation rate) The evolutionary history of the different

modes of energy production will provide some perspective on the selective advantage

which respiration and fermentation conferred as the environmental conditions and

resource constraints changed during the history of life on Earth

The fermentative way of energy generation is now accepted as the primordial mode

of energy processing [33] The invention of photosynthesis changed the atmosphere,

since bacteria and plants used light energy to split water into hydrogen and oxygen

After accumulation of oxygen in the atmosphere, OxPhos as a new way of energy

gen-eration was established by bacteria Such free living bacteria have been integrated into

eukaryotic cells and represent the symbiosis of two different cell types By acquiring

bacteria capable of OxPhos a new organism emerged having the choice between energy

release based on substrate phosphorylation or OxPhos During evolution of higher

ver-tebrates duplication and modification of genes [33] led to a fermentative energy

gen-eration which is not suppressed by oxygen [8] Since this type of glycolysis is

performed even in the presence of oxygen, Warburg called it aerobic glycolysis The

term aerobic glycolysis is somehow misleading since both types of glycolysis are

anae-robic Warburg’s term aerobic glycolysis tries to emphasize the fact that the important

difference to normal glycolysis is that it is not suppressed by oxygen

These observations suggest that in the course of evolution, anaerobic glycolysis arose first The primitive nature of glycolysis, in contrast to OxPhos, is indicated by the fact

that the glycolytic enzymes exist free in solution in the soluble portion of the

cyto-plasm This is in sharp contrast to the enzyme systems responsible for respiration and

photosynthesis These enzymes are grouped and arranged in an intracellular structure

organized in the mitochondria and chloroplasts [20]

Modes of energy processing: their incidence

Most cancer cells have an increased glycolysis However, the relative contribution of

glycolysis to ATP supply varies considerably with the tissue The dependence of

glyco-lytic contribution on tissue type is shown in Table 1[19] Aerobic glycolysis is also

observed in certain normal cells A selected group is given in Table 2[19]

In humans, aerobic glycolysis is extremely important in cells like neurons, retinal cells, stem cells and germ cells Neurons, for example, depend absolutely on glucose as

an energy source and cannot function anaerobically The utility of anaerobic glycolysis,

to a muscle cell for example, when it needs to utilize large amounts of energy in a

short period of time, stems from the fact that the rate of ATP production from

glyco-lysis is approximately 100 times faster than from oxidative phosphorylation During

exertion muscle cells do not need to activate anabolic reaction pathways The

require-ment is to generate the maximum amount of ATP, for muscle contraction, in the

shortest time frame This is the reason why muscle cells derive almost all of the ATP

consumed during exertion from anaerobic glycolysis As these types of cells do not

dis-play the cancer phenotype, it is evident that elevated glycolysis is not a sufficient

con-dition for tumorigenesis

Elevated glycolysis is also not always observed at all stages of tumorigenesis The stu-dies reported in de Groof et al [34] based on transformed fibroblasts in mice, indicate

the multi-step nature of the progression from normalcy to the malignant state These

studies document a high rate of OxPhos in newly transformed cells with a subsequent

high glycolytic rate in the malignant state The experimental system shows that the

proliferation of malignant cells hinges on the evolution of different metabolic profiles

as an adaptive response to the changes in the resource conditions during the transition

from normal to malignant cells

We will now show how these evolutionary changes in metabolic profile can be understood in terms of a measure of selective advantage - a prescription derived from

Quantum Metabolism and evolutionary dynamics

Darwinian fitness in somatic evolution

The concept Darwinian fitness describes the capacity of a variant type to increase in

frequency in competition with an incumbent population for the available resources

This notion is of fundamental importance in all analytical studies of the Darwinian

process at molecular, cellular and organismic levels

Studies of selective advantage in evolutionary processes were pioneered by Fisher [35], who proposed the population growth rate, denoted r, as the measure of Darwinian

fit-ness According to Fisher, selective advantage in evolutionary dynamics is given by

Here, Δr = r* - r, where r* is growth rate of the variant and incumbent type, respec-tively These models have provided qualitative insight into several studies of the

Table 1 Predominant energy metabolism in different types of tumor cells

Tissue of Tumor Cell Type Predominant Energy Metabolism

Colon Colon adenocarcinomas Gly

Lung Lung carcinoma Ox Phos

Skin Melanoma Ox Phos

Table 2 Glycolytic ATP contribution in selected normal cell types

Cell Type Percentage of Glycolytic ATP contribution

Mouse macrophages 18

Rat coronary endothelial cells 53

Human platelets 24

evolutionary process and since 1930 have been the dominant approach in evolutionary

genetics - in spite of several inconsistencies in their predictive and explanatory power

Measure of selective advantage: evolutionary entropy

The analytical and conceptual basis for the limitations of the classical models proposed by

Fisher, was only recently recognized [35] The studies reported in [15] showed that the

population growth rate as a measure of fitness is only valid when population size and

resources are infinite Consequently, the Malthusian parameter will only be a meaningful

selective index when population sizes are large

The studies initiated by Demetrius [15] and later developed as a general model of the evolutionary process showed that the outcome of competition between an incumbent

and a variant type is conditional on the magnitude and the variation in resource

con-straints, and is predicted by the robustness or the demographic stability of the

popula-tion [36] Robustness describes the rate at which the populapopula-tion returns to its steady

state condition after a random perturbation in the individual birth and death rates

This property can be analytically described in terms of the quantity evolutionary

entropy, an information theoretic measure which describes the uncertainty in the state

- age, size, metabolic energy - of the immediate ancestor of a randomly chosen

new-born Evolutionary entropy in cellular populations describes the variability in the rate

at which individual cells pass through the various stages of the cell cycle A

synchro-nous population has small entropy, an asynchrosynchro-nous population has large entropy

Selective advantage in the context of this model is given by

Here,ΔS = S* - S, where S* and S represent the entropy of the variant and the incum-bent type, respectively The quantity F, the reproductive potential, and g, the

demo-graphic index, are statistical parameters which depend on the survivorship and

replication rate of the cells in the population These statistical parameters characterize

certain measures of resource constraints, assuming that the changes in the resource

con-ditions are in dynamical equilibrium with changes in the number of cells The parameter

F describes mean resource abundance and g the variance in the resource abundance

(i) F < 0 corresponds to limited resource (ii) F > 0 corresponds to unlimited resource and

(iii) g < 0 corresponds to a variable resource distribution (iv) g > 0 characterizes a constant resource distribution The measure of selective advantage given by Eq (3) is a far-reaching generalization of the measure given by Eq (2) Indeed, Eq (3) reduces to Eq (2) when g = 0 or when the

population size N tends to infinity The condition g = 0 corresponds to a complete

corre-lation between the resource variability and the demographic variability

In view of the characterization of the statistical parameters F and g as resource con-straints, the measure of selective advantage given by Eq (3) can be roughly described

in terms of the following two tenets

(Ia) When resources are constant and limited, variants with increased entropy will have a selective advantage and increase in frequency in competition with the resident

type